The General Linear Model Questions

2. onsider the model Y; = μ; +ε¡,&; ~ N(0,0²), where µ¡ =α + ßX¡, i = 1,…, n.
a) What is the maximal model and why? Find the maximized log-likelihood under
this model. Call this In.
b) What are the MLE’s of a and B (no deviation necessary)? What is the
maximized log-likelihood. Call this ₁.
c) Ho B=0. What is the maximized log-likelihood? Call this lo.
d) Using results in a), b), and c) develop a test for Ho: B = 0, against HA: B±0.
Obtain the statistic and its distribution.
e) Show that the above test is equivalent to t =
ß
51 √s xx
XX
~
– t(n − 2), wh
–
n
S
XX
§∞ = [(x, − x)³‚ s² = —— ±(v‚ – â – ßx‚)³ .
1
n
–
–
i=1
n-2
i=15. Consider an exponential family of models with canonical parameter 0
f(y; 0,2) = exp[{y0-b(0)}/a(λ) +C(y, λ)].
Now take a sample y₁, y2, . . ., yn. Consider a regression model linking p covariates with
the responses.
a) Derive the mean μ and variance o² of Y.
n
b} = { yixij
b) Derive the sufficient statistic for the regression parameter ẞj, j=1,2, . . ., p.
c) By the chain rule or otherwise obtain the estimating equation for ẞj. Give all details.
d) If ẞk is the estimate of ẞ at the kth iteration using Fisher’s scoring method, write down
the formula for obtaining estimate of ẞ at the (k+1)th iteration, where ẞ is a vector of p
regression parameters, giving detailed expressions for all the terms involved.
14
e) Show that if the sample comes from a normal distribution the Fisher’s scoring method
needs only one iteration.1. Consider the general linear model Y = Xẞ+ €,Є~
N (0,0²1)
c) Suppose in the above model no covariates are involved, i.e., E(Y)=μ. What is
is asymptotically unbiased. Hints: no proof necessary. Use
2 Show that
Elx²(a)]-a and
02
(n-1)s²
-X² (n-1).